Pure future local temporal logics are

Abstract not found

. It is shown that it is decidable whether an equation over a free partially commutative monoid has a solution. We give a proof of this result using normal forms. Our method is a direct reduction of a trace equation system to a word equation system with regular constraints. Hereby we use the extension of Makanin's theorem on the decidability of word equations to word equations with regular constraints, which is due to Schulz. Keywords: word equations, partial commutations, traces, word equations with regular constraints 1 Introduction Solving equations is a central topic in various elds of computer science, especially concerning unication, as required by automated theorem proving or logic programming. A famous result of Makanin [15] states that there exists an algorithm deciding for a given equation L = R, where L; R 2 ( [ ) contain both unknowns from and constants from , whether an assignment : ! exists satisfying (L) = (R). In a more general setting it is k...

Abstract. Recently, local logics for Mazurkiewicz traces are of increasing interest. This is mainly due to the fact that the satisfiability problem has the same complexity as in the word case. If we focus on a purely local interpretation of formulae at vertices (or events) of a trace, then the satisfiability problem of linear temporal logics over traces turns out to be PSPACE–complete. But now the difficult problem is to obtain expressive completeness results with respect to first order logic. The main result of the paper shows such an expressive completeness result, if the underlying dependence alphabet is a cograph, i.e., if all traces are series parallel posets. Moreover, we show that this is the best we can expect in our setting: If the dependence alphabet is not a cograph, then we cannot express all first order properties.

Abstract. We summarize several characterizations, inclusions, and separations on fragments of first-order logic over words and Mazurkiewicz traces. The results concerning Mazurkiewicz traces can be seen as generalizations of those for words. It turns out that over traces it is crucial, how easy concurrency can be expressed. Since there is no concurrency in words, this distinction does not occur there. In general, the possibility of expressing concurrency also increases the complexity of the satisfiability problem. In the last section we prove an algebraic and a language theoretic characterization of the fragment Σ2[E] over traces. Over words the relation E is simply the order of the positions. The algebraic characterization yields decidability of the membership problem for this fragment. For words this result is well-known, but although our proof works in a more general setting it is quite simple and direct. An essential step in the proof consists of showing that every homomorphism from a free monoid to a finite aperiodic monoid M admits a factorization forest of finite height. We include a simple proof that the height is bounded by 3 |M|. 1

We review some results on global and local temporal logic on Mazurkiewicz traces. Our main contribution is to show how to derive the expressive completeness of global temporal logic with respect to first order logic [9] from the similar result on local temporal logic [11].

. We consider some basic problems on the decidability and complexity of trace rewriting systems. The new contribution of this paper is an O(n log(n)) algorithm for some computing irreducible normal forms in the case of certain one-rule systems. 1 Introduction The notes of this paper are based on the Font Romeu Lecture and on an invited lecture at FCT-93 conference [11] of the second author. In the first part of the paper we give some basic background about trace rewriting systems. There is some overlap with the published notes from FCT-93. However, the second part is original and constitutes a new contribution to the theory of trace rewriting systems. The theory of rewriting over free partially commutative monoids (trace rewriting) combines combinatorial aspects from string rewriting (modulo a congruence) and graph rewriting. This restriction of graph rewriting leads to feasible algorithms, but some interesting complexity questions are still open. For example, a challenging open probl...

We show that the existential theory of free partially commutative monoids with involution is decidable. As a consequence the existential theory of graph groups is also decidable. If the underlying alphabet of generators is fixed, we obtain a Pspace-completeness result, otherwise (in the uniform setting) our decision procedure is in ExpSpace. Our proof is a reduction to the main result of [6].

Linear-time algorithms are presented for several problems concerning words in a partially commutative monoid, including whether one word is a factor of another and whether two words are conjugate in the monoid. tfl 1990 Academic Press. Inc. 1.